From a Riemann–Hilbert perspective, in the presence of Fisher–Hartwig singularities, one has to construct the \(4\times 4\)  local parametrices near the points \(z_i\) . Expectedly, these local parametrices must be expressed in terms of confluent hypergeometric functions as suggested by [1]}. We have not yet worked out the details of this construction but we believe that it should be well within reach. It would be methodologically important to achieve the results obtained from operator-theoretic tools via the Riemann–Hilbert approach as well. Moreover, we expect that the evenness of the Fisher–Hartwig part of \(\phi \)  would not play a role in our \(4 \times 4\)  setting, and in that sense there are reasonable prospects of generalizing the results of [2]} to symbols \(\phi \)  with non-even Fisher–Hartwig part.
